Geography Reference
In-Depth Information
We begin with a general set-up of the mapping equations for
case
(
i
) circular cylinder and
case
(
ii
) circular cone by
x
y
=
U
(23.55)
f
(
V
)
x
y
=
r
cos
α
or α
=
nU, r
=
g
(
V
)
(23.56)
sin
α
,
r) which cover
R
2
.
Case
(
i
):
The mapping equations (
23.55
) have been set-up in such a way that the
circular cylinder
inter-
sects the
circular paraboloid
at
V
=1, the vertical coordinate, namely for
case
(
i
1), a
conformal
mapping.
In contrast for
case
(
i
2), an
equiareal mapping,
and
case
(
i
3), an
equidistant map-
ping.
For
V
=0
,y
=
f
(0)=0 is required which
g
enerates—as we shall see—a cylinder of radius
either in terms of Cartesian
coordinates
(
x, y
)or
polar coordinates
(
α
f
(1) = (5
√
5
−
1)
/
12 for
case
(
i
2) and
f
(1) =
√
5
/
2+arcsinh2
/
4for
case
(
i
3), respectively.
Case
(
ii
): Let us denote the
conic angle θ
such that
n
=sin
θ
and
α
=
U
sin
θ
hold for the
conic constant
and the polar coordinate, the
azimuth,
respectively. The mapping equations (
23.56
) have been set-
up in such a way that for
V
=0
,x
=
y
= 0 is an identity. In particular, for
case
(
ii1
), a
conformal
mapping,
the
circular cone
intersects the
circular paraboloid
at
V
=1, the vertical coordinate. In
contrast, for
case
(
ii2
), an
equiareal mapping,
and
case
(
ii3
), an
equidistant mapping,
for
V
=0,
r
=
g
(0)
=
0
is postulated which gene
ra
tes—as we shall see—a cone of radius g(1) =
5
√
5
1
/
√
6
n
for
case
(
ii2
) and g(1) =
√
5
/
2+arcsinh2
/
4for
case
(
ii
3), respectively.
−
M
2
of type paraboloid
P
2
, vertical section, projective design
Fig. 23.15.
Two-dimensional Riemann manifold
C
1
of radius one (normal placement) and of the circular cone
C
0
(normal placement)
of the circular cylinder
Thus we are left with the problem to
determine the unknown function f
(
V
)for
case
(
i
)aswell
as
g
(
V
)for
case
(
ii
). Here we follow the
constructive approach
of the
theory of map projections
in cases where the structure of the map (
23.55
)aswellas(
23.56
)isgiven.Its
generic steps
are
outlined in Boxes
23.17
-
23.20
:The
first generic step
is based upon the “
left Jacobi map
”being
represented by the matrices (
23.58
)and(
23.59
)of
first derivatives
subject to the
condition
(
23.60
)
which preserves the orientation of the differential map (
dx, dy
)or(
dα, dr
)
→
(
dU
,
dV
) also called
“
pullback
”. The representation of the “
left Cauchy-Green deformation tensor
”
C
l
:=
J
l
G
r
J
l
subjects to the “
right metric tensor
”
G
r
=
I
2
(unit matrix) for
case
(
i
)or
G
r
=
Diag
(
r
2
,
1) for
case
(
ii
), respectively. As soon as we compare the “
left metric tensor
”
G
l
of the paraboloid
P
2
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